fluids Article Statistics of Extreme Waves in Coastal Waters: Large Scale Experiments and Advanced Numerical Simulations Jie Zhang 1,2, Michel Benoit 1,2,* , Olivier Kimmoun 1,2, Amin Chabchoub 3,4 and Hung-Chu Hsu 5 1 École Centrale Marseille, 13013 Marseille, France; [email protected] (J.Z.); [email protected] (O.K.) 2 Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE UMR 7342, 13013 Marseille, France 3 Centre for Wind, Waves and Water, School of Civil Engineering, The University of Sydney, Sydney, NSW 2006, Australia; [email protected] 4 Marine Studies Institute, The University of Sydney, Sydney, NSW 2006, Australia 5 Department of Marine Environment and Engineering, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan; [email protected] * Correspondence: [email protected] Received: 7 February 2019; Accepted: 20 May 2019; Published: 29 May 2019 Abstract: The formation mechanism of extreme waves in the coastal areas is still an open contemporary problem in fluid mechanics and ocean engineering. Previous studies have shown that the transition of water depth from a deeper to a shallower zone increases the occurrence probability of large waves. Indeed, more efforts are required to improve the understanding of extreme wave statistics variations in such conditions. To achieve this goal, large scale experiments of unidirectional irregular waves propagating over a variable bottom profile considering different transition water depths were performed. The validation of two highly nonlinear numerical models was performed for one representative case. The collected data were examined and interpreted by using spectral or bispectral analysis as well as statistical analysis. The higher probability of occurrence of large waves was confirmed by the statistical distributions built from the measured free surface elevation time series as well as by the local maximum values of skewness and kurtosis around the end of the slope. Strong second-order nonlinear effects were highlighted as waves propagate into the shallower region. A significant amount of wave energy was transmitted to low-frequency modes. Based on the experimental data, we conclude that the formation of extreme waves is mainly related to the second-order effect, which is also responsible for the generation of long waves. It is shown that higher-order nonlinearities are negligible in these sets of experiments. Several existing models for wave height distributions were compared and analysed. It appears that the generalised Boccotti’s distribution can predict the exceedance of large wave heights with good confidence. Keywords: coastal areas; extreme waves; statistical analysis; bispectral analysis; nonlinear wave models 1. Introduction Extreme wave, also known as freak wave or rogue wave, refers in oceanography to large water wave with crest-to-trough wave height H exceeding twice the significant wave height Hs in the wave field, or with wave crest height hc higher than 1.25Hs [1]. In a Gaussian sea state, wave heights H follow a Rayleigh distribution when the wave field is assumed to be narrow-banded. In such cases, large waves fulfilling the criteria H/Hs > 2 are not so unusual, occurring approximately once every 3000 waves. For instance, if the average wave period Tave = 15 s, it implies that the observer could Fluids 2019, 4, 99; doi:10.3390/fluids4020099 www.mdpi.com/journal/fluids Fluids 2019, 4, 99 2 of 24 probably encounter one of such waves in 12 h. What makes this particular field of research interesting is the fact that the extreme waves may occur not only in the energetic storm sea state but also in a calm sea state, making these waves outstanding and exceptional compared to the surrounding waves. These extreme waves are supposed to be very rare basing on Rayleigh distribution model, whereas they seem to have a larger occurrence in the real world, as suggested in the following scientific studies and reports [2–5]. The possible mechanisms of the formation of extreme waves are summarised and discussed in recent reviews [6,7]. One of the well-known mechanisms of the generation of freak waves is the so-called Benjamin–Feir (or modulational) instability [8,9], which is frequently studied within the framework of the nonlinear Schrödinger equation (NLS). The NLS equation can describe the evolution of the narrow-banded weakly nonlinear waves [10–13]. More recent reviews show that real sea states are more complex and one needs to consider three-dimensionality [14], dissipation [15–17] and breaking [18] to achieve more accurate modeling of such wave fields. For constant water depth, the modulation instability of unidirectional waves should disappear with the relative water depth kh < 1.363 (k denotes the wave number and h is the water depth) [8,19]. However, for the multidirectional sea states, this is no longer the case. The three-dimensionality not only affects the modulation instability but also the statistical parameters [20,21]; the sea states are either focusing or defocusing depending on the spreading angle. Although we are aware of the essential importance of short-crest waves, the topic of this paper is restricted to one horizontal propagation direction case since the uneven bottom effects on freak waves are not fully understood yet in such configuration. Recently, it has been emphasised that the formation of extreme waves is more related to the second/third-order non-resonant or bound harmonic waves than modulation instability, especially in the finite water depth case where the instabilities are further attenuated [22]. The evidence of the link between a water depth transition and a higher probability of the occurrence of extreme waves has been shown both experimentally [23,24], and numerically [25–27]. There are also studies dealing with extreme wave statistics in coastal areas with different shapes of variable bathymetry. Katsardi et al. (2013) [28] conducted experimental tests with unidirectional waves propagating over mild bed slopes (1:100 and 1:250) including breaking zones, and made extensive comparisons of wave height distributions. Nonlinear transformation of irregular waves propagating over sloping bottoms (1:15, 1:30 and 1:45) is discussed using wavelet-based bicoherence in [29]. Ma et al. (2014) [30] studied spatial variations of skewness, kurtosis and groupiness factor for irregular waves over a bar (1:20) in shallow water. The obliquity effects on the statistical parameters are discussed in [21] using 3D simulations with a High-Order Spectral (HOS) model. In this context, the objectives of the present study were three-fold: (1) studying experimentally the statistical distribution of (extreme) wave heights considering variable bottom coastal conditions in a uni-directional water wave flume; (2) comparing the collected temporal wave profiles and corresponding statistical distributions with the results from two advanced numerical models; and (3) comparing the measured and simulated wave height distributions with a number of existing statistical models. Regarding Objective 1, the spatial evolution of unidirectional irregular sea states due to the presence of a constant bottom gradient was studied in the large scale facility of Tainan Hydraulics Laboratory (THL) in Taiwan. This large facility allowed us to investigate the full life cycle of extreme waves, from the generation to the degeneration. Several cases with different experimental conditions were tested, while one representative case was investigated for the validation of numerical nonlinear models. In addition to the overall evolution of the sea state, particular attention was paid to the sloping bottom zone as well as the two depth transition regions. Regarding Objective 2, two accurate numerical models were adopted to simulate the experiments, both based on the fully nonlinear potential flow theory: a higher-order Boussinesq-type model based on the model in [31] and the highly nonlinear and dispersive model, whispers3D, using a spectral approximation of the potential in the vertical direction [32,33]. As shown below, the corresponding results indicate that the increase of the probability of occurrence of freak waves was clearly observed in experiments and Fluids 2019, 4, 99 3 of 24 well captured in the simulations, over a region where the water depth is relatively mildly varying. Both measured and simulated wave profiles were studied by using statistical, spectral and bispectral analysis. Finally, Objective 3 was devoted to the statistical distributions of measured and simulated wave heights comparing with six existing statistical distribution models. The present article is structured as follows. In Section2, the experimental set-up and the tested sea state conditions are introduced. The mathematical models and their numerical implementations are presented in Section3. Several signal processing techniques are adopted to interpret the obtained data: from the measurements as well as the simulations, including spectral analysis (Section4), bispectral analysis (Section5), nonlinear wave parameters analysis (Section6), and statistical analysis of the wave height distribution (Section7). Conclusively, the main results are summarised in Section8. 2. Experimental Set-Up and Test Conditions The experiments were conducted in the 2-D wave flume of THL at Tainan, Taiwan. The flume is 200 m long and 2 m wide. The waves were initially generated in the deeper region with depths of h = 1.2 m and 1.3 m. The bathymetry was decomposed into three parts: a 30 m long flat part, a 20 m long part with a constant slope of 1/20 and again a 120 m long flat bottom part. Waves were generated at x = 0 m by a piston-type wave-maker, which was also able to absorb part of the wave energy reflected by the uneven bottom or far-field rip-rap mound. The last 30 m part of the flume worked as an effective wave absorber including a deep water area and a mound of stones placed at the end of the flume. Over the flume length, 30 capacitance gauges were used to measure the wave elevation with a sampling frequency of 100 Hz.